A173119 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3, read by rows.

1, 1, 1, 1, 5, 1, 1, 6, 6, 1, 1, 7, 12, 7, 1, 1, 8, 28, 28, 8, 1, 1, 9, 36, 56, 36, 9, 1, 1, 10, 45, 119, 119, 45, 10, 1, 1, 11, 55, 164, 238, 164, 55, 11, 1, 1, 12, 66, 219, 483, 483, 219, 66, 12, 1, 1, 13, 78, 285, 702, 966, 702, 285, 78, 13, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 3.

Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 3. - G. C. Greubel, Apr 27 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  5,  1;
  1,  6,  6,   1;
  1,  7, 12,   7,   1;
  1,  8, 28,  28,   8,   1;
  1,  9, 36,  56,  36,   9,   1;
  1, 10, 45, 119, 119,  45,  10,   1;
  1, 11, 55, 164, 238, 164,  55,  11,  1;
  1, 12, 66, 219, 483, 483, 219,  66, 12,  1;
  1, 13, 78, 285, 702, 966, 702, 285, 78, 13, 1;

Mathematica

T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j, 0, 5}]];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)

Prog

(Sage)
@CachedFunction
def T(n, k, q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Apr 27 2021

Crossrefs

Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), this sequence (q=3), A173120 (q= -4), A173122.

Sequence in context: A280374 A259975 A028313 * A050178 A297986 A298635

Adjacent sequences: A173116 A173117 A173118 * A173120 A173121 A173122

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 10 2010

Extensions

Edited by G. C. Greubel, Apr 27 2021

Status

approved